Hypothesis Testing for Population Means and Proportions

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• Hypothesis Testing for Population Means and
  Proportions

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Topics

• Hypothesis testing for population means
• z test for the simple case (in last lecture)
• z test for large samples
• t test for small samples for normal distributions
• Hypothesis testing for population proportions
• z test for large samples

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z-test for Large Sample Tests

• We have previously assumed that the population
  standard deviationsis known in the simple case.
• In general, we do not know the population
  standard deviation, so we estimate its value with
  the standard deviation s from an SRS of the
  population.
• When the sample size is large, the z tests are
  easily modified to yield valid test procedures
  without requiring either a normal population or
  known s.
• The rule of thumb n gt 40 will again be used to
  characterize a large sample size.

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z-test for Large Sample Tests (Cont.)

• Test statistic
• Rejection regions and P-values
• The same as in the simple case
• Determination of ß and the necessary sample size
• Step I Specifying a plausible value of s
• Step II Use the simple case formulas, plug in
  thes estimation for step I.

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t-test for Small Sample Normal Distribution

• z-tests are justified for large sample tests by
  the fact that A large n implies that the sample
  standard deviation s will be close tosfor most
  samples.
• For small samples, s and sare not that close any
  more. So z-tests are not valid any more.
• Let X1,., Xn be a simple random sample from N(µ,
  s). µ and s are both unknown, andµ is the
  parameter of interest.
• The standardized variable

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The t Distribution

• Facts about the t distribution
• Different distribution for different sample sizes
• Density curve for any t distribution is symmetric
  about 0 and bell-shaped
• Spread of the t distribution decreases as the
  degrees of freedom of the distribution increase
• Similar to the standard normal density curve, but
  t distribution has fatter tails
• Asymptotically, t distribution is
  indistinguishable from standard normal
  distribution

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Table A.5 Critical Values for t Distributions
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t-test for Small Sample Normal Distribution
(Cont.)

• To test the hypothesis H0µ µ0 based on an SRS
  of size n, compute t test statistic
• When H0 is true, the test statistic T has a t
  distribution with n -1 df.
• The rejection regions and P-values for the t
  tests can be obtained similarly as for the
  previous cases.

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Recap Population Proportion

• Let p be the proportion of successes in a
  population. A random sample of size n is
  selected, and X is the number of successes in
  the sample.
• Suppose n is small relative to the population
  size, then X can be regarded as a binomial random
  variable with

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Recap Population Proportion (Cont.)

• We use the sample proportion as an
  estimator of the population proportion.
• We have
• Hence is an unbiased estimator of the
  population proportion.

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Recap Population Proportion (Cont.)

• When n is large, is approximately normal.
  Thus
• is approximately standard normal.
• We can use this z statistic to carry out
  hypotheses for
• H0 p p0 against one of the following
  alternative hypotheses
• Ha p gt p0
• Ha p lt p0
• Ha p ? p0

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Large Sample z-test for a Population Proportion

• The null hypothesis H0 p p0
• The test statistic is

Alternative Hypothesis P-value Rejection Region for Level a Test
Ha p gt p0 P(Z z) z za
Ha p lt p0 P(Z z) z - za
Ha p ? p0 2P(Z z ) z za/2
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Determination of ß

• To calculate the probability of a Type II error,
  suppose that H0 is not true and that p p ?
  instead. Then Z still has approximately a normal
  distribution but
• ,
• The probability of a Type II error can be
  computed by using the given mean and variance to
  standardize and then referring to the standard
  normal cdf.

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Determination of the Sample Size

• If it is desired that the level atest also have
  ß(p?) ß for a specified value of ß, this
  equation can be solved for the necessary n as in
  population mean tests.